Polynomial Functions of Higher Degree

2.2 Polynomial Functions of Higher Degree

What is a polynomial?

1. The value of "n" must be a non-negative integer.(can't be negative and a whole number)

    2. The coefficients are real numbers.

    3.  The degree is the highest value "n" without "a" = 0

The possibilities of polynomial graphs:

The graphs of polynomials all follow two basic principles:

     1. They are continuous( i.e can be drawn without lifting pencil)

     2. Have smooth rounded curves

             Polynomial                                           Not a polynomial

Continuous, Curvy/Smooth

Not Continuous, Not smooth

How to sketch a polynomial(without plotting points):

Step 1. Find the zeros.

The rules for sketching goes as follows:

     1. The zero are where the graph touches the x-axis

            a. If the zero is repeated an odd amount of times, it passes through

            b. If the zero is repeated an even amount of times, it is tangent to the axis and "bounces" off the line

     2. The term with the highest exponent matters

           a. Coefficient

                   i. If this is positive the graph increases traveling towards positive infinity

                  ii. If this is negative the graph decreases traveling towards positive infinity

           b. Exponent

                  i. If the exponent is odd, the positive and negative sides of the graph approach their respective                          infinities going the opposite direction vertically

                  ii.  If the exponent is even, the positive and negative sides of the graph approach their respective                        infinities going the same direction vertically

The reasoning for this is that while approaching infinity, the term with the highest exponent overwhelms the other terms and dictates the entire graph.

The leading coefficient was positive, the degree was odd, and the root 2 was repeated once